introduction to factoring polynomials section 10.4
TRANSCRIPT
Introduction to Factoring Introduction to Factoring PolynomialsPolynomials
Section 10.4
Finding the Greatest Common FactorFinding the Greatest Common Factor
The first step in factoring a polynomial is to see whether the terms of the polynomial have a common factor.
If there is a common factor, we can write the polynomial as a product by factoring factoring outout the common factor.
We will usually factor out the greatest common factor (GCF).
2Martin-Gay, Prealgebra, 5ed
Finding the Greatest Common Factor. . .Finding the Greatest Common Factor. . .
The greatest common factor (GCF) of a list greatest common factor (GCF) of a list of termsof terms is the product of the GCF of the numerical coefficients and the GCF of the variable factors.
Consider the terms, and .15 2 2x y 6 3x y
15 3 52 2x y x x y y
6 2 33x y x x x y
Common factors denoted by
3 x x yGCF 3 2x y3
Helpful Hint
Notice below that the GCF of a list of terms contains the smallest exponent on each common variable.
The GCF of , , and is x y5 6 x y2 7 x y3 4
x y2 4Smallest exponent on x.
Smallest exponent on y.
4Martin-Gay, Prealgebra, 5ed
Factoring Out the Greatest Common FactorFactoring Out the Greatest Common Factor
• Do the terms have a greatest common factor other than 1?
To factor a polynomial:
• If so, factor out the greatest common factor from each term by writing each term as a product of the greatest common factor and the term’s remaining factors.
• Use the distributive property to write the factored form of the polynomial.
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Factoring Out the Greatest Common Factor Factoring Out the Greatest Common Factor
• The GCF of 5x + 10 is 5.
Consider, 5x + 10
• Factor 5 from each term and write each term as a product of 5 and the remaining terms,
• Using the distributive property, write
5x + 10 5 5 2x
5 2( )xfactored form of polynomialfactored form of polynomial
Factoring canbe checked by
multiplying.
Factoring canbe checked by
multiplying.
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Helpful Hint
A factored form of 5x + 10 is not
5 5 2 x
Although the termsterms have been factored (written as a product), the polynomialpolynomial 5x + 10 has not been factored. A factored form of 5x + 10 is the productproduct 5(x + 2).
5 5 2 x
5(x + 2)
factored termsfactored terms
factored polynomialfactored polynomial7
Examples of Factored PolynomialsExamples of Factored Polynomials
x x3 5 x x3 21e jDon’t forget the 1.Don’t forget the 1.
8 10 24 2a a a 2 4 5 13a a a e jDon’t forget the -1.Don’t forget the -1.
8 10 24 2a a a 2 4 5 13a a ae jNotice the changes in signs when factoring -2aNotice the changes in signs when factoring -2a
In this example,factor out -2arather than 2a
In this example,factor out -2arather than 2a
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